- Open Access
Standing waves and global existence for nonlinear wave equations with potential, strong, and nonlinear damping terms
© Huang; licensee Springer 2014
- Received: 27 October 2013
- Accepted: 27 May 2014
- Published: 10 September 2014
This paper is concerned with the Cauchy problem of nonlinear wave equations with potential, strong, and nonlinear damping terms. Firstly, by using variational calculus and compactness lemma, the existence of standing waves of the ground states is obtained. Then the instability of the standing wave is shown by applying potential-well arguments and concavity methods. Finally, we show how small the initial data are for the global solutions to exist.
- wave equations
- nonlinear damping terms
- strong damping terms
- global existence
For the case of linear damping (, ) and nonlinear sources, Levine  showed that the solutions to (1.1) with negative initial energy blow-up for the abstract version. For the nonlinear damping and source terms (, , , ), the abstract version has been considered by many researchers –. For instance, Georgiev and Todorova  prove that if , a global weak solution exists for any initial data, while if the solution blows up in finite time when the initial energy is sufficiently negative. In , Todorova considers the additional restriction on . Ikehata  considers the solutions of (1.1) with small positive initial energy, using the so-called ‘potential-well’ theory. The case of strong damping (, , ) and nonlinear source terms () has been studied by Gazzola and Squassina in . They prove the global existence of solutions with initial data in the potential well and show that every global solution is uniformly bounded in the natural phase space. Moreover, they prove finite time blow-up for solutions with high energy initial data. However, they do not consider the case of a nonlinear damping term (, , ).
For simplicity, throughout this paper we denote by and arbitrary positive constants by .
From Proposition 2.1, it follows that is the critical case, namely for , a weak solution exists globally in time for any compactly supported initial data; while for , blow-up of the solution to the Cauchy problem (1.1) occurs.
The main results of this paper are the following.
There exists such that
(a2) is a ground state solution of (1.4).
From Theorem 2.3, we have the following.
(b1) If, and there existssuch that, then the solutionof the Cauchy problem (1.1) blows up in a finite time.
In this section, we prove Theorem 2.3.
This completes the proof of Lemma 3.1. □
is bounded below on M and.
Next, we solve the variational problem (2.6).
We first give a compactness lemma in .
Letwhenandwhen. Then the embeddingis compact.
In the following, we prove Theorem 2.3.
Proof of Theorem 2.3
Next we prove (a2) of Theorem 2.3.
So far, we have completed the proof of Theorem 2.3. □
In this section, we prove Theorem 2.5.
Let, thenandare invariant under the flow generated by (1.1).
Similarly, we can show that is also invariant under the flow generated by (1.1). This completes the proof of Lemma 4.1. □
Next, we prove Theorem 2.5.
Proof of Theorem 2.5
By (2.7), we have .
Next, we prove (b2) of Theorem 2.5.
Thus, we complete the proof of Theorem 2.5. □
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